# Equation of Astroid/Cartesian Form

## Theorem

Let $H$ be the astroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 4 b$.

Let $C_2$ be embedded in a cartesian coordinate plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.

Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The point $P = \tuple {x, y}$ is described by the equation:

- $x^{2/3} + y^{2/3} = a^{2/3}$

## Proof

By definition, an astroid is a hypocycloid with $4$ cusps.

From the parametric form of the equation of an astroid, $H$ can be expressed as:

- $\begin{cases} x & = 4 b \cos^3 \theta = a \cos^3 \theta \\ y & = 4 b \sin^3 \theta = a \sin^3 \theta \end{cases}$

Squaring, taking cube roots and adding:

\(\displaystyle x^{2/3} + y^{2/3}\) | \(=\) | \(\displaystyle a^{2/3} \paren {\cos^2 \theta + \sin^2 \theta}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a^{2/3}\) | Sum of Squares of Sine and Cosine |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Hypocycloid with Four Cusps: $11.8$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid